Related papers: On some random walks driven by spread-out measures

We consider the model of directed polymers in a random environment introduced by Petermann : the random walk is $\mathbb{R}^d$-valued and has independent gaussian $N(0,I_d)$-increments, and the random media is a stationary centred Gaussian…

Recent years have witnessed much progress on Gaussian and bootstrap approximations to the distribution of sums of independent random vectors with dimension $d$ large relative to the sample size $n$. However, for any number of moments $m>2$…

Let $X_1,X_2,...$ be independent identically distributed random variables with values in $\C$. Denote by $\mu$ the probability distribution of $X_1$. Consider a random polynomial $P_n(z)=(z-X_1)...(z-X_n)$. We prove a conjecture of Pemantle…

We consider an extended version of Horn's problem: given two orbits $\mathcal{O}_\alpha$ and $\mathcal{O}_\beta$ of a linear representation of a compact Lie group, let $A\in \mathcal{O}_\alpha$, $B\in \mathcal{O}_\beta$ be independent and…

The classical Dvoretzky--Rogers lemma provides a deterministic algorithm by which, from any set of isotropic vectors in Euclidean $d$-space, one can select a subset of $d$ vectors whose determinant is not too small. Subsequently,…

We study the asymptotic behavior of the long cycles of a random permutation of $n$ objects with respect to multiplicative measures with polynomial growing cycle weights. We show that the longest cycle and the length differences between the…

We present a survey of results related to the Milnor's problem on group growth. We discuss the cases of polynomial growth, exponential but not uniformly exponential growth, but the main part of the article is devoted to the intermediate…

Let $T \colon M \to M$ be a nonuniformly expanding dynamical system, such as logistic or intermittent map. Let $v \colon M \to \mathbb{R}^d$ be an observable and $v_n = \sum_{k=0}^{n-1} v \circ T^k$ denote the Birkhoff sums. Given a…

A connection is made between the random turns model of vicious walkers and random permutations indexed by their increasing subsequences. Consequently the scaled distribution of the maximum displacements in a particular asymmeteric version…

We consider a discrete time random walk in one dimension. At each time step the walker jumps by a random distance, independent from step to step, drawn from an arbitrary symmetric density function. We show that the expected positive maximum…

This paper develops a general framework for analyzing asymptotics of $V$-statistics. Previous literature on limiting distribution mainly focuses on the cases when $n \to \infty$ with fixed kernel size $k$. Under some regularity conditions,…

A closed equilateral random walk in 3-space is a selection of unit length vectors giving the steps of the walk conditioned on the assumption that the sum of the vectors is zero. The sample space of such walks with $n$ edges is the…

We explore the asymptotic distributions of sequences of integer-valued additive functions defined on the symmetric group endowed with the Ewens probability measure as the order of the group increases. Applying the method of factorial…

We derive explicit formulae for the expected volume and the expected number of facets of the convex hull of several multidimensional Gaussian random walks in terms of the Gaussian persistence probabilities. Special cases include the already…

Several new geometric quantile-based measures for multivariate dispersion, skewness, kurtosis, and spherical asymmetry are defined. These measures differ from existing measures, which use volumes and are easy to calculate. Some theoretical…

This paper proves a generalization of a conjecture of Guoniu Han, inspired originally by an identity of Nekrasov and Okounkov. The main result states that certain sums over partitions p of n, involving symmetric functions of the squares of…

In this work, we achieve gap amplification for the Small-Set Expansion problem. Specifically, we show that an instance of the Small-Set Expansion Problem with completeness $\epsilon$ and soundness $\frac{1}{2}$ is at least as difficult as…

The object under consideration in this article is the total volume $V_{g,n}(x_1, \ldots, x_n)$ of the moduli space of hyperbolic surfaces of genus $g$ with $n$ boundary components of lengths $x_1, \ldots, x_n$, for the Weil-Petersson volume…

We study the errors brought by finite volume effects and dilution effects on the practical determination of the count probability distribution function P_N(n,L), which is the probability of having N objects in a cell of volume L^3 for a set…

For words of length $n$, generated by independent geometric random variables, we consider the mean and variance of the number of inversions and of a parameter of Knuth from permutation in situ. In this way, $q$--analogues for these…